标题： 从一个框架曲线构造有界平面区域的等距浸入 显示英文标题

标题： Construction of an isometric immersion of a bounded, planar region from a framed curve

摘要： 我们开发了一个框架，用于表征简单连通、有界、边界为分段光滑的平面区域到三维空间中的等距浸入。 每个浸入都与图像曲面边界上的一个带框架的曲线相关联，该曲线由参数化曲线和单位法向量组成。 我们确定了该带框架的曲线上的一组相容性和正则性条件，这些条件确保存在一个$C^1$的等距浸入，在几乎处处为$C^2$并且具有有限弯曲能量。 在这些条件下，我们推导出弯曲能量到边界曲线上的线积分的精确维度约简，而无需依赖渐近假设或近似。 通过分析框架边界上单位法向量的行为，我们区分了浸入曲面的平面区域和曲面区域。 我们确定了全局$C^2$正则性可能丢失的几何条件，在这种情况下，相关的浸入属于$W^{2,2}$—— 一种在不可拉伸弹性曲面的变分模型中自然出现的 Sobolev 空间。 显示英文摘要

摘要： We develop a framework for characterizing isometric immersions of simply connected, bounded, planar regions with piecewise smooth boundaries into three-dimensional space. Each immersion is associated with a framed curve along the boundary of the image surface, comprised by a parametrized curve and a unit normal vector. We identify a set of compatibility and regularity conditions on this framed curve that ensure the existence of a $C^1$ isometric immersion that is $C^2$ almost everywhere and possesses finite bending energy. Under these conditions, we derive an exact dimensional reduction of the bending energy to a line integral over the boundary curve, without relying on asymptotic assumptions or approximations. By analyzing the behavior of the unit normal vector along the framed boundary, we distinguish between planar and curved regions of the immersed surface. We identify the geometric conditions under which global $C^2$ regularity is potentially lost, in which case the associated immersion belongs to $W^{2,2}$ -- a Sobolev space that arises naturally in variational models of unstretchable elastic surfaces.